lecture11 notes

lecture11 notes - LECTURE 11 Last time The channel coding...

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LECTURE 11 Last time: The channel coding theorem overview Upper bound on the error probability Bound on not being typical Bound on too many elements being typ- ical Coding theorem (weak) Lecture outline Strong coding theorem Revisiting channel and codes Bound on probability of error Error exponent
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Revisiting channel and codes Consider a DMC with transition probabili- ties P Y X ( y x ) | | For any block length N , let N P Y N | X N ( y | x N ) = i N =1 P Y | X ( y i | x i ) P X N ( x N ) = N i =1 P X ( x i ) P Y N ( y N ) = i n =1 P Y ( y i ) in particular we can select the input prob- ability to be capacity-achieving, since IID inputs yield capacity for a DMC the output alphabet Y and the input alpha- bet X may be different
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Revisiting channel and codes The code is a block code with bit blocks of length L being mapped onto code se- quences of length N For binary sequences, the block code maps all the possible M = 2 L binary sequences onto sequences x N The rate of a block code is R = log M N Let τ c be the duration of an output symbol y from Y , the data rate in bits is R τ c For any positive integer and R > 0, a ( N, R ) block code is a code of length N which has 2 NR codewords
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Upper bound on probability Recall that for the weak coding theorem we performed a typicality-based decoding That decoding led to a WLLN type of ar- gument, which was the source of the poor handle we have on the behavior of error probability with N Let us then consider another criterion for decoding: maximum likelihood
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